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The digits in parentheses are the uncertainty (standard deviation) in the last two digits of the measured value.
The universal gas constant R is simply the Boltzmann constant multiplied by Avogadro's number. The gas constant is more useful when calculating numbers of particles in moles.
Given a thermodynamic system at an absolute temperature T, the Boltzmann constant defines an energy E = kT that is, roughly speaking, the typical amount of thermal energy carried by each microscopic particle in the system. For example, an atom in a classical ideal gas has a mean kinetic energy of 1.5 kT. The energy kT associated with room temperature, 300 K (27 °C, or 80 °F), is 4.14 × 10^{21} J (25.9 meV).
In statistical mechanics, the entropy S of a system is defined as the natural logarithm of Ω, the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy E):
The constant of proportionality k is the Boltzmann constant. This equation, which relates the microscopic details of the system (via Ω) to its macroscopic state (via the entropy S), is the central idea of statistical mechanics.
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